Structure results for higher order symmetry algebras of 2D classical superintegrable systems

TL;DR

This paper extends a recurrence relation method to analyze the structure of higher order symmetry algebras in 2D classical superintegrable systems, including several notable models, providing new insights into their algebraic properties.

Contribution

The authors adapt their quantum superintegrability method to classical systems and explore the structure of their symmetry algebras across multiple models, including systems on the 2-sphere.

Findings

Constructed classical analogs of quantum symmetry structures

Applied method to five classical systems including the caged oscillator

Suggested symmetries are of lowest possible order in most cases

Abstract

Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the associated classical Hamiltonian systems, as well as for a family including the generic potential on the 2-sphere. We do not have a proof in every case that the generating symmetries are of lowest possible order, but we believe this to be so via an extension of our method.